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- Title
An arithmetic Hilbert-Samuel theorem for pointed stable curves.
- Authors
i Montplet, Gerard Freixas
- Abstract
Let .O;6; F1 be an arithmetic ring of Krull dimension at most 1, S D SpecO and .X SI 1n/a pointed stable curve. Write U D X n S j j .S. For every integer 0, the invertible sheaf kC1 X=S.k1 Cn inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface U1. In this article we define a Quillen type metric kkQ on the determinant line kC1 D.C1 X=S.1n and compute the arithmetic degree of .kC1; kkQ by means of an analogue of the Riemann-Roch theorem in Arakelov geometry. As a byproduct, we obtain an arithmetic Hilbert-Samuel formula: the arithmetic degree of .-kC1; k - k L2 / admits an asymptotic expansion in k, whose leading coefficient is given by the arithmetic self-intersection of .!X=S.-1 C - - - C -n/; k - khyp/. Here k - k L2 and k - khyp denote the L2 metric and the dual of the hyperbolic metric, respectively. Examples of application are given for pointed stable curves of genus 0.
- Subjects
MATHEMATICS theorems; ARAKELOV theory; RIEMANN-Roch theorems; RIEMANN surfaces; INTEGERS
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2012, Vol 14, Issue 2, p321
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/JEMS/304